Rutgers/Lucent ALLIESIN TEACHING MATHEMATICSAND TECHNOLOGY Grant 2000Using technology not simply to do things better, but to do better things.

USING THE GEOMETER'S SKETCHPAD TO EXPLORE MIDPOINT QUADRILATERALS

I will begin by assuming that most of you have had some exposure to the Geometer's Sketchpad (GSP) before. As a review, show that you can:

create a non-special quadrilateral;

change the labels of the vertices;

measure its angles, and show that the sum of the angles remain constant even as the shape of the figure is dynamically changed;

construct the interior of the figure, color it red, and measure the area;

display a table that shows the perimeter and area for different quadrilaterals.

To illustrate how the Sketchpad can be used to explore geometric relationships, follow the sequence of steps outlined below:

Begin with a new sketch. In Display >> Preferences, turn off Autoshow Labels for all objects.

Construct another non-special quadrilateral.

Construct the midpoints of all of the sides; connect the midpoints to form a "midpoint quadrilateral". Observe the shape of the midpoint quadrilateral as, in turn, you grab-&-drag each of the vertices.

Make as many conjectures as you can about this inner figure.

Describe how you could confirm or deny the correctness of each conjecture.

For example, one conjecture is that the inner figure is a parallelogram. You could confirm or deny that by measuring the slopes of the sides of both figures. If it's true, what reasonable explanation might there be for this?

Exploration: Assuming the inner figure is always a parallelogram, under what conditions is the inner figure a rectangle? a rhombus? a square?